A hereditarily indecomposable Banach space with rich spreading model structure

Abstract

We present a reflexive Banach space $$\mathfrak{X}_{usm}$$ which is Hereditarily Indecomposable and satisfies the following properties. In every subspace Y of $$\mathfrak{X}_{usm}$$ there exists a weakly null normalized sequence {y n } n , such that every subsymmetric sequence {z n } n is isomorphically generated as a spreading model of a subsequence of {y n } n . Also, in every block subspace Y of $$\mathfrak{X}_{usm}$$ there exists a seminormalized block sequence {z n } and $$T:\mathfrak{X}_{usm} \to \mathfrak{X}_{usm}$$ an isomorphism such that for every n ∈ ℕ, T(z 2n−1) = z 2n . Thus the space is an example of an HI space which is not tight by range in a strong sense.